A363126 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k non-modes, all 0's removed.

1, 1, 2, 3, 4, 1, 4, 3, 8, 3, 6, 8, 1, 10, 9, 3, 11, 13, 6, 15, 18, 9, 13, 24, 18, 1, 25, 24, 25, 3, 19, 36, 40, 6, 29, 41, 52, 13, 33, 45, 79, 19, 42, 57, 95, 36, 1, 39, 68, 133, 54, 3, 62, 72, 158, 87, 6, 55, 87, 214, 121, 13, 81, 95, 250, 177, 24

Offset: 0

Author

Gus Wiseman, May 16 2023

Keywords

• nonn
• tabf

Comments

A non-mode in a multiset is an element that appears fewer times than at least one of the others. For example, the non-modes in {a,a,b,b,b,c,d,d,d} are {a,c}.

Examples

			Triangle begins:
   1
   1
   2
   3
   4   1
   4   3
   8   3
   6   8   1
  10   9   3
  11  13   6
  15  18   9
  13  24  18   1
  25  24  25   3
  19  36  40   6
  29  41  52  13
  33  45  79  19
  42  57  95  36   1
  39  68 133  54   3
Row n = 9 counts the following partitions:
  (9)          (441)       (3321)
  (54)         (522)       (4221)
  (63)         (711)       (4311)
  (72)         (3222)      (5211)
  (81)         (6111)      (42111)
  (333)        (22221)     (321111)
  (432)        (32211)
  (531)        (33111)
  (621)        (51111)
  (222111)     (411111)
  (111111111)  (2211111)
               (3111111)
               (21111111)
		

Crossrefs

Row sums are A000041.

Row lengths are approximately A000196.

Column k = 0 is A047966.

For modes we have A362614, rank statistic A362611.

For co-modes we have A362615, rank statistic A362613.

Columns k > 1 sum to A363124.

Column k = 1 is A363125.

This rank statistic (number of non-modes) is A363127.

For non-co-modes we have A363130, rank statistic A363131.

A008284/A058398 count partitions by length/mean.

A275870 counts collapsible partitions.

A353836 counts partitions by number of distinct run-sums.

A359893 counts partitions by median.

Cf. A002865, A053263, A098859, A237984, A238478, A327472, A353863, A353864, A362612.

Programs

Mathematica

nmsi[ms_]:=Select[Union[ms],Count[ms,#]